To solve these problems, we'll evaluate the given function values based on the provided graph description of the functions $f$ and $g$. Let's go step by step:

a. $f(g(6))$

1. Find $g(6)$:
   According to the description, the line $g$ passes through the point $(6, 3)$. So, $g(6) = 3$.

2. Find $f(3)$:
   From the description of the curve for $f$, it passes through the point $(3, 4)$. Hence, $f(3) = 4$.

Thus, $f(g(6)) = f(3) = 4$.

b. $g(f(-1))$

1. Find $f(-1)$:
   From the graph description, the function $f$ passes through $(-1, 4)$, so $f(-1) = 4$.

2. Find $g(4)$:
   Since there is no direct information about $g(4)$ in the description, we need to assume based on the line's behavior or additional points if available. As no specific $g(4)$ point is given, we may infer that this value could lie between existing points if interpolation is required. However, without exact interpolation details, this part is incomplete.

c. $f(f(2))$

1. Find $f(2)$:
   The description does not provide the exact value for $f(2)$. However, we know that the curve passes through points $(1, 0)$ and $(3, 4)$, so we can assume that $f(2)$ lies between those values. Given the curve's trend, it seems that $f(2) = 2$ might be a reasonable interpolation.

2. Find $f(2)$:
   From the assumption above, $f(2) = 2$.

Thus, $f(f(2)) = f(2) = 2$.

d. $g(g(3))$

1. Find $g(3)$:
   From the graph description, $g(3) = 0$.

2. Find $g(0)$:
   From the graph, the value of $g(0)$ is not directly given, but based on the points $(2, -2)$, $(3, 0)$, and extrapolating the linear trend, we can reasonably assume that $g(0) = -1$.

Thus, $g(g(3)) = g(0) = -1$.

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Final Answers:

a. $f(g(6)) = 4$
b. $g(f(-1)) =$ Incomplete, requires $g(4)$
c. $f(f(2)) = 2$
d. $g(g(3)) = -1$

Would you like further clarification on any part, or would you like me to provide more detail on how interpolation works for these problems?

Here are 5 questions related to this concept:

1. How do you interpret composite functions graphically?
2. What are typical methods for estimating values between points on a graph?
3. How does linear interpolation work for line graphs like $g(x)$?
4. How do you calculate the value of a function at non-given points on a graph?
5. What strategies can you use to solve composite functions when some values are unknown?

Tip: When working with graphs, estimating values between known points using trends like linear or curve fitting is a useful strategy when exact values are not provided.